Persistence Paths and Signature Features in Topological Data Analysis

We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations—barcode to path, path to tensor series—results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.